Constructing Conflict-Free Schedules via the Affine Plane AG(2,5)

This paper studies a scheduling problem related to organizing twenty-five students into six separate work sessions. In each session, the students are divided into five groups, with five students in each group. One important condition in this problem is that any two students are allowed to work together only once during all sessions. This condition helps ensure fairness and prevents repeated grouping. To solve this problem, the affine plane of order five, , is used as a mathematical model. The affine plane is constructed algebraically from the projective plane over the finite field . In the proposed model, students are represented by points, while work groups are represented by lines. The use of parallel classes in makes it possible to organize the groups without conflicts, so that no student pair is repeated. Although this study focuses on the case of twenty-five students, the results indicate that the combination of finite geometry and companion matrix methods can also be useful for solving similar scheduling and allocation problems.